Download Descriptive Set Theory and Dimension

Descriptive Set Theory
and Dimension
Taras Banakh
(Lviv, Ukraine — Kielce, Poland)
7 July, 2009
All spaces are separable and metrizable.
Standard Notation for Borel Classes:
Π0
α is the multiplicative class,
Σ0
α is the additive class
of absolute Borel spaces corresponding to a
countable ordinal α.
0
The classes Π0
α , Σα are dual each to the other.
In particular:
Π0
1 is the class of compact spaces,
Σ0
2 is the class of σ-compact spaces,
Π0
2 is the class of Polish spaces,
Π0
3 is the class of absolute Fσδ -sets.
Motivation:
Theorem (Wadge, Louveau, Saint Raymond)
0
Let Γ ∈ {Π0
ξ , Σξ : ξ ≥ 3} be a Borel class and
Γ̌ be its dual.
For a space X̃ ∈ Γ̌ and a Borel subset X ⊂ X̃
TFAE:
0) X ∈
/ Γ̌;
0’) X̃ \ X ∈
/ Γ;
1) For any 0-dimensional space C ∈ Γ there is
a closed embedding f : C ,→ X;
2) For any subspace C ∈ Γ of a 0-dimensional
compact space K there is an embedding e :
K ,→ X̃ with e−1(X) = C;
3) For a subspace C ∈ Γ of a 0-dimensional
compact space K there is a map e : K ,→ X̃
with e−1(X) = C;
4) For a 0-dimensional space C ∈ Γ there is a
perfect map f : C → X.
A map f : X → Y is perfect if the preimage f −1 (K) of
any compact K ⊂ Y is compact.
Problem: Do (1)–(4) remain equivalent if one
replaces “0-dimensional” with “n-dimensional”?
Some Terminology
Definition 1: Let Γ be a class of spaces. A
space X is called
• Γ-universal if for each space C ∈ Γ there is a
closed embedding f : C ,→ X;
• Γ-preuniversal if each space C ∈ Γ there is a
perfect map f : C ,→ X.
Similar notions can be introduced for pairs of spaces
(X̃, X) where X ⊂ X̃.
~ be a class of pairs. A pair
Definition 2: Let Γ
(X̃, X) is called
~
~ there
• Γ-universal
if for each pair (K, C) ∈ Γ
is a closed embedding f : K ,→ X̃ such that
f −1(X) = C;
~
~ there
• Γ-preuniversal
if each pair (K, C) ∈ Γ
is a perfect map f : K → X̃ with f −1(X) = C.
For two classes of spaces K, C let
(K, C) = {(K, C) : K ∈ K, C ∈ C}.
For a class Γ of spaces, let
Γ[n] = {C ∈ Γ : dim C ≤ n}.
In particular, Π0
1 [n] will stand for the class of
metrizable compact spaces of dimension ≤ n.
Theorem (Wadge, Louveau, Saint Raymond)
(expanded version)
0 : ξ ≥ 3} be a Borel class and Γ̌
Let Γ ∈ {Π0
,
Σ
ξ
ξ
be its dual Borel class. For a space X̃ ∈ Γ̌ and
a Borel subspace X ∈ X̃ TFAE:
0) X ∈
/ Γ̌;
0’) X̃ \ X ∈
/ Γ;
1) X is Γ[0]-universal;
1’) X̃ \ X is Γ̌[0]-universal;
2) X is Γ[0]-preuniversal;
2’) X̃ \ X is Γ̌[0]-preuniversal;
3) (X̃, X) is (Π0
1 [0], Γ)-universal;
3) (X̃, X̃ \ X) is (Π01 [0], Γ̌)-universal;
4) (X̃, X) is (Π0
1 [0], Γ)-preuniversal.
4’) (X̃, X̃ \ X) is (Π01 [0], Γ̌)-preuniversal.
Problem: What about higher-dimensional version of this theorem?
Theorem (Banakh, 1996):
0 : ξ ≥ 3} be a Borel class,
Let Γ ∈ {Π0
,
Σ
ξ
ξ
Γ̌ be its dual Borel class, and n ∈ ω ∪ {∞}. For
a Polish space X̃ and a subspace X ∈ X̃ TFAE:
1) X is Γ[n]-universal;
1’) X̃ \ X is Γ̌[n]-universal;
2) X is Γ[n]-preuniversal;
3’) X̃ \ X is Γ̌[n]-preuniversal;
4) (X̃, X) is (Π0
1 [n], Γ)-universal;
4’) (X̃, X̃ \ X) is (Π01 [n], Γ̌)-universal;
5) (X̃, X) is (Π0
1 [n], Γ)-preuniversal.
5’) (X̃, X̃ \ X) is (Π01 [n], Γ̌)-preuniversal.
Open Problem: Is this theorem true if X̃ ∈ Γ̌?
(Yes, if n = 0 and X is Borel).
This theorem is a partial case of a more general result
treating the universality for sufficiently rich classes of
pairs.
~ of pairs is:
Definition: A class Γ
• compact if for each pair (K, C) ∈ Γ the space
K is compact;
~ ⇒ (2ω ×K, 2ω ×C) ∈ Γ;
~
• 2ω -stable if (K, C) ∈ Γ
~ and each
• Fσ -additive if for each (K, C) ∈ Γ
~
Fσ -subset A ⊂ K we get (K, C ∪ A) ∈ Γ;
~ and
• Gδ -multiplicative if for each (K, C) ∈ Γ
~
each Gδ -subset G ⊂ K we get (K, G ∩ A) ∈ Γ.
Remark: For any Borel class Γ ∈ {Π0ξ , Σ0ξ : ξ ≥ 3} and
any n ∈ {ω ∪ {∞} the class (Π01 [n], Γ) is compact, 2ω stable, Fσ -additive, and Gδ -multiplicative.
Theorem (Banakh, 1996):
Given a compact 2ω -stable Fσ -additive
~ let
Gδ -multiplicative class of pairs Γ
~ and
Γ = {C : (K, C) ∈ Γ}
~
Γ̌ = {K \ C : (K, C) ∈ Γ}.
For a Polish space X̃ and a subspace X ∈ X̃
TFAE:
1) X is Γ-universal;
1’) X̃ \ X is Γ̌-universal;
2) X is Γ-preuniversal;
3’) X̃ \ X is Γ̌-preuniversal;
~
4) (X̃, X) is Γ-universal;
~
5) (X̃, X) is Γ-preuniversal.
Some corollaries
~ Γ, and Γ̌ from the preceding theFor classes Γ,
orem and a Γ-universal space X ∈ Γ any perfect continuous image Y of X contains a closed
topological copy of X and hence dim Y ≥ dim X.
So, the dimension of Γ-universal spaces cannot
be lowered by perfect maps.
In particular, any perfect image of the n-dimensional
Nöbeling space (which is Π0
2 [n]-universal) has
dimension ≥ n.
Constructions increasing the Borel complexity
1. Spaces of measures
Theorem (??, Banakh-Radul, 1997)
Let PR (X) be the space of probability Radon measures
on X, endowed with the weak-star topology.
If X is Σ0
<ξ [0]-universal, then
PR (X) is Π1
ξ [∞]-universal.
Theorem (??, Banakh-Cauty, 1997)
Let Pβ (X) be the space of probability measures with
compact support on X.
If X is Σ1
n [0]-universal, then
Pβ (X) is Π1
n+1 [∞]-universal.
2. Hyperspaces
Theorem (??, Banakh, 1997)
Let X be a connected locally path-connected space and
2X be the hyperspace of non-empty compact subsets of
X, endowed with the Vietoris topology.
If X is Σ1
n [1]-universal, then
2X is Π1
n+1 [∞]-universal.
3. Countable powers
Trivial Observation:
1) The interval I = [0, 1] is a 1-dimensional
ω
compact space with Π0
1 [∞]-universal power I .
2) The real line R is a Polish 1-dimensional
ω
having Π0
1 [∞]-universal countable power R .
What about higher Borel classes?
Theorem (Banakh-Cauty, 2001):
1) For any finite-dimensional space X the countable power X ω is not Σ0
2 [s.c.d]-universal, where
Σ02 [s.c.d] is the class of spaces that are countable unions
of finite-dimensional compacta;
2) For any countable-dimensional space X the
power X ω is not Σ0
3 -universal.
4. Finite powers
Theorem (Menger-Nöbeling-Pontryagin-Lefschetz)
If X is contains an arc, then
X 2n+1 is Π0
1 [n]-universal.
Theorem (Banakh-Cauty-Truščak-Zdomsky, 2006)
If X is a locally path-connected nowhere locally
compact space, then X n+1 is Π0
2 [n]-universal.
Theorem (Banakh-Cauty, 2005)
If X is a meager locally path-connected space,
then X 2n+1 is Σ0
2 [n]-universal.
Theorem (Bowers, 1985; Banakh-Cauty, 2005)
0 , Σ0 } there is a 1For a Borel class Γ ∈ {Π0
,
Π
1
2
2
dimensional space X ∈ Γ such that each power
X n+1 is Γ[n]-universal.
Such a space X is a subspace of a dendrite with dense
set of end-points.
Open Problem. Which Borel classes Γ do
contain a 1-dimensional space X with Γ[n]universal powers X n+1?
References
T.Banakh, T.Radul, M.Zarichnyi, Absorbing sets in infinitedimensional manifolds // Mat. Stud. Monograph Series. 1), VNTL Publishers, Lviv, 1996. 240p.
T.Banakh, R.Cauty, Interplay between strongly universal spaces and pairs // Dissert. Math. 286 (2000)
1–38.
T.Banakh, R.Cauty, On universality of countable and
weak products of sigma hereditarily disconnected spaces
// Fund. Math. 167 (2001) 97–109.
T.Banakh, R.Cauty, Kh.Trushchak, L.Zdomskyy, On
universality of finite products of Polish spaces // Tsukuba
J. Math. 28:2 (2004) 455-471.
T.Banakh, R.Cauty, On universality of finite powers of
locally path connected meager spaces // Colloquium
Math. 102:1 (2005), 87-95.
T.Banakh, T.Radul, Topology of spaces of probability
measures // Matem. Sbornik. 188 (1997) 23–46.
T.Banakh, R.Cauty, On hyperspaces of nowhere topologically complete spaces // Mat. Zametki. 62:1 (1997),
35–51.
T.Banakh, Topological classification of spaces of probability measures on projective spaces // Mat. Zametki.
61 (1997), 441–444.
Thank You!